It's the prosecutorial service who make the decision on whether or not to prosecute, not the police (or indeed, the victim).
When they are making a decision, they consider whether the evidential threshold has been met. So in cases where a victim of assault has indicated that they wish to withdraw their statement, the prosecutorial service will consider if there is any other, independent evidence, such as other witnesses, CCTV, medical evidence etc.
If the evidential test has been met, they will then consider the public interest test. In a case where the victim is indicating they do not wish to proceed, they consider aspects such as severity of the injuries, previous offending history etc. The state has a positive obligation under article 2 ECHR to protect life, and this can include bringing a prosecution against the complainant's wishes.
I have seen cases where an offender has been successfully prosecuted when the victim had withdrawn her evidence, but it was very difficult to achieve. A particular case which stands out had lots of witnesses, and the circumstances were horrific. The victim's husband had previously tried to murder her, and she had withdrawn her complaint in that previous matter, and because there had only been 2 of them in the house at the time, the prosecution couldn't go ahead. On the most recent occasion the husband knocked her unconscious by punching her in the face repeatedly in a maternity ward, whilst holding a 6 hour old baby, in front of all the other new mothers and staff. 
He completely intimidated her, and she withdrew the complaint, but the prosecution went ahead nonetheless. At court she claimed she and her H were happily reconciled, and there had been no further incidents, however police were able to produce evidence that she was living in a refuge, and had been there every night. He was eventually jailed, and couldn't believe it as he thought he had successfully intimidated his wife into perjuring herself.
In cases such as this, it is very difficult to achieve a prosecution though.